Maximum Bracket Subsequence

AtCoder

IDagc071_b

Time2000ms

Memory256MB

Difficulty—

You are given integers $N$, $K$, and a correct bracket sequence $S$ of length $K$ consisting of `(` and `)`. Find the number, modulo $998244353$, of length-$N$ strings $T$ consisting of `(` and `)` that satisfy all of the following conditions: * There exists a (not necessarily contiguous) subsequence of $T$ that is a correct bracket sequence of length $K$. * Among all (not necessarily contiguous) subsequences of $T$ that are correct bracket sequences of length $K$, the lexicographically largest one is $S$. Here, concerning the lexicographical order of strings consisting of `(` and `)`, we consider `(` smaller than `)`. Notes on correct bracket sequences A correct bracket sequence is a string that can be transformed into the empty string by repeatedly deleting a substring that is `()` zero or more times. Notes on lexicographical orderA string $S = S_1S_2\ldots S_{|S|}$ is said to be **lexicographically smaller** than a string $T = T_1T_2\ldots T_{|T|}$ if either of the following holds. 1. $|S| < |T|$ and $S_1S_2\ldots S_{|S|} = T_1T_2\ldots T_{|S|}$. 2. There exists an integer $1 \leq i \leq \min\lbrace |S|, |T| \rbrace$ such that: * $S_1S_2\ldots S_{i-1} = T_1T_2\ldots T_{i-1}$, * $S_i$ is a character smaller than $T_i$. ## Constraints * $K \leq N \leq 10^7$ * $2 \leq K \leq 5 \times 10^5$ * $N$ and $K$ are integers. * $S$ is a correct bracket sequence of length $K$. ## Input The input is given from Standard Input in the following format: $N$ $K$ $S$ [samples]

Samples

Input #1

7 4
(())

Output #1

20

For example, if $T=$`((())))`, the only (not necessarily contiguous) subsequence of $T$ that is a correct bracket sequence of length $4$ is `(())`, which equals $S$, so it satisfies the conditions.
If $T=$`((())()`, there are two (not necessarily contiguous) subsequences of $T$ that are correct bracket sequences of length $4$: `()()` and `(())`. The lexicographically largest one is `()()`, so it does not satisfy the conditions.

Input #2

20 4
()()

Output #2

Input #3

71 10
()(()())()

Output #3

190654464

Input #4

10000000 28
(()()(()))(()((()))())()(())

Output #4

769035381

API Response (JSON)

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    "name": "Maximum Bracket Subsequence",
    "description": {
      "content": "You are given integers $N$, $K$, and a correct bracket sequence $S$ of length $K$ consisting of `(` and `)`. Find the number, modulo $998244353$, of length-$N$ strings $T$ consisting of `(` and `)` th",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
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    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc071_b"
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      "statement_type": "Markdown",
      "content": "You are given integers $N$, $K$, and a correct bracket sequence $S$ of length $K$ consisting of `(` and `)`.\nFind the number, modulo $998244353$, of length-$N$ strings $T$ consisting of `(` and `)` th...",
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